KAM trapping and recurrent cell crossing in ABC flows

Fix an initial value $z(0)=z_0$. The $xy$-plane is divided into disjoint KAM and non-KAM regions, and a KAM region is one of the regions in which the associated trajectories are trapped within a cell. KAM-trapping conjecture. Any trajectory that begins in the KAM region will remain trapped in the cell it begins in, although it need not remain in the KAM region. Any trajectory that begins outside a KAM region will cross cell boundaries infinitely many times. This conjecture combines trapping in KAM regions with recurrent cell-boundary crossing in the non-KAM region; the source presents it as numerically supported and does not give a proof or resolution.